Data

times = tacdata$Time/60
reftac <- tacdata$RefCBL
roitac <- tacdata$FSLSSTR
weights <- tacdata$Weights

First all the data is loaded (unshown). This data includes:

• Times
  • Middle of each frame in minutes
• Reference TAC
  • Cerebellar Grey Matter
• ROI TAC
  • Striatum
• Weights
  • Range: 0:1

This data is plotted below:

par(mfrow=c(3,1))

plot(times, reftac)
plot(times, roitac)
plot(times, weights)

Reference Region Model Data

SRTM

\[C(t) = R_{1}C_{R}(t)+(k_{2}-\frac{R_{1}k_{2}}{1+BP_{ND}})C_{R}(t)\otimes e^{-\frac{k_{2}}{1+BP_{ND}}t}\]

SRTM requires input of the times, reftac and roitac. Weights are optional.

srtmout <- srtm(times, reftac, roitac, weights)

pandoc.table(srtmout$par, digits = 3)

Plot

The output object can be used to visualise the fit:

plot_srtmfit(srtmout)

SRTM Fit Check

… and the residuals:

plot_residuals(srtmout)

SRTM Residuals

pandoc.table(summary(srtmout))

pandoc.table(glance(srtmout$fit), digits=3)

Non-Invasive Logan Plot

refLogan requires input of the times, reftac, roitac and k2prime. Weights are optional.

k2prime can be obtained from the srtm fit:

k2prime = srtmout$par$k2 / srtmout$par$R1
k2prime

## [1] 0.1292453

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

taclow = tacdata$FSLSFC    # Frontal Cortex 
tacmed = tacdata$fslLST    # Limbic Striatum
tachigh = tacdata$fslAST   # Associative Striatum

refLogan_tstar(times, reftac, taclow, tacmed, tachigh, k2prime = k2prime, 'demonstration')

## png 
##   2

Using this information, we see that we should probably use 9 included frames for the fit.

refloganout <- refLogan(times, reftac, roitac, k2prime, 
                        tstarIncludedFrames = 9, weights=weights)

pandoc.table(refloganout$par)

Plot

The output object can be used to visualise the fit:

plot_refLoganfit(refloganout)

RefLogan Fit Check

… and the residuals:

plot_residuals(refloganout)

RefLogan Residuals

pandoc.table(summary(refloganout))

pandoc.table(glance(refloganout$fit), digits = 3)

MRTM1

MRTM1 requires input of the times, reftac and roitac. Weights are optional.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

mrtm1_tstar(times, reftac, taclow, tacmed, tachigh, 'demonstration')

Using this information (unshown), we see that we should probably use all 14 frames for the fit.

mrtm1out <- mrtm1(times, reftac, roitac, tstarIncludedFrames = 14, weights=weights)

pandoc.table(mrtm1out$par, digits = 3)

Plot

The output object can be used to visualise the fit:

plot_mrtm1fit(mrtm1out)

MRTM1 Fit Check

… and the residuals:

plot_residuals(mrtm1out)

MRTM1 Residuals

pandoc.table(summary(mrtm1out))

pandoc.table(glance(mrtm1out$fit), digits = 3)

MRTM2

MRTM2 requires input of the times, reftac, roitac and k2prime. Weights are optional.

k2prime can be obtained from the mrtm1 fit:

k2prime = mrtm1out$par$k2prime
k2prime

## [1] 0.1295072

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

mrtm2_tstar(times, reftac, taclow, tacmed, tachigh, k2prime=k2prime, 'demonstration')

Using this information (unshown), we see that we should probably use all 14 frames for the fit.

mrtm2out <- mrtm2(times, reftac, roitac, k2prime = k2prime, 
                  tstarIncludedFrames = 14, weights=weights)

pandoc.table(mrtm2out$par)

Plot

The output object can be used to visualise the fit:

plot_mrtm2fit(mrtm2out)

MRTM2 Fit Check

… and the residuals:

plot_residuals(mrtm2out)

MRTM2 Residuals

pandoc.table(summary(mrtm2out))

pandoc.table(glance(mrtm2out$fit), digits = 3)

Non-Invasive Multilinear Logan Plot

\[\int _{0} ^{t} C_{T} (\tau) d\tau = DVR\Bigg(\int _{0} ^{t} C_{T}' (\tau) d\tau+\frac{C_{T}'(t)}{k2'}\Bigg)-bC_{T}(t)\]

refmlLogan is the kinetic model used in WAPI, which is detailed in Turkheimer’s 2003 IEEE paper. This model requires input of the times, reftac and roitac and k2prime. Weights are optional.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

taclow = tacdata$FSLSFC    # Frontal Cortex 
tacmed = tacdata$fslLST    # Limbic Striatum
tachigh = tacdata$fslAST   # Associative Striatum

refmlLogan_tstar(times, reftac, taclow, tacmed, tachigh, k2prime, 'demonstration')

Using this information (unshown), we see that we should probably use 9 included frames for the fit. We shall use 6.

refmlloganout <- refmlLogan(times, reftac, roitac, k2prime,
                        tstarIncludedFrames = 9, weights=weights)

pandoc.table(refmlloganout$par, digits = 3)

Plot

The output object can be used to visualise the fit:

plot_refmlLoganfit(refmlloganout)

RefmlLogan Fit Check

… and the residuals:

plot_residuals(refmlloganout)

RefLogan Residuals

pandoc.table(summary(refmlloganout))

pandoc.table(glance(refmlloganout$fit), digits = 3)

Patlak Reference Tissue Model

refPatlak requires input of the times, reftac and roitac. Weights are optional.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

taclow = tacdata$FSLSFC    # Frontal Cortex 
tacmed = tacdata$fslLST    # Limbic Striatum
tachigh = tacdata$fslAST   # Associative Striatum
refPatlak_tstar(times, reftac, taclow, tacmed, tachigh, 'demonstration')

Since this is actually a reversible tracer, we shouldn’t really be using refPatlak at all, but we can use four frames to test it out (unshown).

refpatlakout <- refPatlak(times, reftac, roitac, 
                        tstarIncludedFrames = 4, weights=weights)

pandoc.table(refpatlakout$par, digits=4)

## 
## --------
##    K    
## --------
## 0.005227
## --------

Plot

The output object can be used to visualise the fit:

plot_refPatlakfit(refpatlakout)

RefPatlak Fit Check

… and the residuals:

plot_residuals(refpatlakout)

RefPatlak Residuals

pandoc.table(summary(refpatlakout))

pandoc.table(glance(refpatlakout$fit), digits = 3)

Data

First all the data is loaded (unshown). This data includes:

• TAC Times
  • Middle of each frame in minutes
• ROI TAC
  • Grey Matter
• Weights
  • Range: 0:1
• Blood TACs
• Plasma TACs
• Blood Times
  • Minutes
• Parent Fraction
• The Delay (can be fitted, or inputted)

The data is plotted below:

par(mfrow=c(2,2))

plot(t_tac, tac)
plot(blooddata$Time.sec., blooddata$Cbl.nCi.cc., xlab = 'Time (sec)', ylab = 'Cb')
plot(plasmadata$Time.sec., plasmadata$Cpl.nCi.cc., xlab = 'Time (sec)', ylab = 'Cp')
plot(parentdata$Times, parentdata$Fraction, xlab = 'Time (sec)', ylab = 'Parent Fraction')

Compartmental Model Data

First, we must organise the blood data and put it together. This function can also accept parent-fraction-corrected plasma data too.

input <- blood_interp(t_blood = blooddata$Time.sec./60, 
                      blood = blooddata$Cbl.nCi.cc.,
                      t_plasma = plasmadata$Time.sec./60, 
                      plasma = plasmadata$Cpl.nCi.cc.,
                      t_parentfrac = parentdata$Times/60,
                      parentfrac = parentdata$Fraction)

One-Tissue Compartment Model

1TCM requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function and vB are also optional. If they are not included, then the function will fit them.

onetcmout <- onetcm(t_tac, tac, input, weights=weights)

pandoc.table(onetcmout$par, digits = 3)

Input Delay

The inpshift is the fitted delay. We can visualise the matching of the two curves without any time shifting (i.e. as they are in the same time scale):

plot_inptac_timings(t_tac, tac, input, inpshift = 0)

Delay Check Before

The timing of these two clearly do not match. Now we can visualise the fit with the time shifting fitted by the one tissue compartment model: in this case, the delay is negative, which means that the TAC was shifted to be a little bit later.

plot_inptac_timings(t_tac, tac, input, inpshift = onetcmout$par$inpshift)

Delay Check After

This parameter has clearly performed a good job in fitting the timing of the two curves to one another.

Plot

The output object can be used to visualise the model fit:

plot_1tcmfit(onetcmout)

1TCM Fit Check

… and the residuals:

plot_residuals(onetcmout)

1TCM Residuals

pandoc.table(summary(onetcmout))

pandoc.table(glance(onetcmout$fit), digits=3)

Two-Tissue Compartment Model

2TCM requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function and vB are also optional. If they are not included, then the function will fit them. In this case, we can use the delay from the 1tcm for the 2tcm fit, and it will not be fitted.

twotcmout <- twotcm(t_tac, tac, input, weights=weights, inpshift = onetcmout$par$inpshift)

pandoc.table(twotcmout$par, digits = 3)

Plot

The output object can be used to visualise the model fit:

plot_2tcmfit(twotcmout)

2TCM Fit Check

… and the residuals:

plot_residuals(twotcmout)

2TCM Residuals

pandoc.table(summary(twotcmout))

pandoc.table(glance(twotcmout$fit), digits=3)

Logan Plot

The Logan Plot method requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function and vB are also optional. If they are not included, then they will be set to zero. For the input delay, we can use the delay from the 1tcm fit. If vB is included, it will be corrected for before the parameter estimation using the following formula:

\[ C_{T}(t) = \frac{C_{Measured}(t) - vB\times C_{B}(t)}{1-vB} \]

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

lowroi = inptacdata$PUT
medroi = inptacdata$GM
highroi = inptacdata$gmCER

Logan_tstar(t_tac, lowroi, medroi, highroi, input, filename='demonstration', 
            inpshift = onetcmout$par$inpshift, vB = 0.05, gridbreaks=4)

Using this information (unshown), we see that we should probably use 10 frames for the fit.

loganout <- Loganplot(t_tac, tac, input, 10, weights, inpshift = onetcmout$par$inpshift, vB=0.05)

loganout$par

##         Vt
## 1 4.225652

Plot

The output object can be used to visualise the model fit:

plot_Loganfit(loganout)

Logan Plot Fit Check

… and the residuals:

plot_residuals(loganout)

Logan Plot Residuals

pandoc.table(summary(loganout))

pandoc.table(glance(loganout$fit), digits=3)

MA1

The MA1 method requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function is also optional. If it is not included, then it will be set to zero.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

lowroi = inptacdata$PUT
medroi = inptacdata$GM
highroi = inptacdata$gmCER

ma1_tstar(t_tac, lowroi, medroi, highroi, input, filename='demonstration', 
          inpshift = onetcmout$par$inpshift, gridbreaks=4)

## png 
##   2

Using this information, we see that we should probably use very few frames for the fit: 11 seems reasonably good and reasonably stable.

ma1out <- ma1(t_tac, tac, input, 11, weights, inpshift = onetcmout$par$inpshift)

ma1out$par

##         Vt
## 1 4.143303

Plot

The output object can be used to visualise the model fit:

plot_ma1fit(ma1out)

MA1 Plot Fit Check

… and the residuals:

plot_residuals(ma1out)

MA1 Plot Residuals

pandoc.table(summary(ma1out))

pandoc.table(glance(ma1out$fit), digits=3)

MA2

The MA2 method requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function is also optional. If it is not included, then it will be set to zero.

ma2out <- ma2(t_tac, tac, input, weights, inpshift = onetcmout$par$inpshift)

pandoc.table(ma2out$par, digits = 3)

Plot

The output object can be used to visualise the model fit:

plot_ma2fit(ma2out)

MA2 Plot Fit Check

… and the residuals:

plot_residuals(ma2out)

MA2 Plot Residuals

pandoc.table(summary(ma2out))

pandoc.table(glance(ma2out$fit), digits=3)

Multilinear Logan Plot

\[\int _{0} ^{t} C_{T} (\tau) d\tau = V_{T}\int _{0} ^{t} C_{P} (\tau) d\tau-\frac{1}{K_{2}}C_{T}(t)\]

The multilinear Logan Plot is the kinetic model used in WAPI, which is detailed in Turkheimer’s 2003 IEEE paper. This model requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function is also optional, and if it is not included, it will be set to zero.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

lowroi = inptacdata$PUT
medroi = inptacdata$GM
highroi = inptacdata$gmCER

mlLogan_tstar(t_tac, lowroi, medroi, highroi, input, filename='demonstration', 
            inpshift = onetcmout$par$inpshift, gridbreaks=4)

Using this information (unshown), we see that we should probably use 10 frames for the fit.

mlloganout <- mlLoganplot(t_tac, tac, input, 10, weights, inpshift = onetcmout$par$inpshift)

pandoc.table(mlloganout$par, digits=3)

Plot

The output object can be used to visualise the model fit:

plot_mlLoganfit(mlloganout)

mlLogan Plot Fit Check

… and the residuals:

plot_residuals(mlloganout)

mlLogan Plot Residuals

pandoc.table(summary(mlloganout))

pandoc.table(glance(mlloganout$fit), digits=3)

Patlak Plot

The Patlak Plot method requires input of the TAC times, TAC and input. Weights are optional. The time shift of the input function is also optional. If it is not included, then it will be set to zero. For the input delay, we can use the delay from the 1tcm fit.

First, we wish to ascertain what a suitable t* value would be. For this, we use TACs from a low-, medium- and high-binding region.

Patlak_tstar(t_tac, lowroi, medroi, highroi, input, filename='demonstration', 
            inpshift = onetcmout$par$inpshift, gridbreaks=4)

Since this is actually a reversible tracer, we shouldn’t really be using the Patlak Plot at all, but we can use nine frames to test it out (unshown).

patlakout <- Patlakplot(t_tac, tac, input, 9, weights, inpshift = onetcmout$par$inpshift)

pandoc.table(patlakout$par, digits=3)

Plot

The output object can be used to visualise the model fit:

plot_Patlakfit(patlakout)

Patlak Plot Fit Check

… and the residuals:

plot_residuals(patlakout)

Patlak Plot Residuals

pandoc.table(summary(patlakout))

pandoc.table(glance(patlakout$fit), digits=3)

SIME

The SIME method requires input of the TAC times, several TACs, input and a grid of V_ND values. Weights are optional. ROI sizes can optionally be included to weight the contributions of the different ROIs. The time shift of the input function and vB are also optional. If they are not included, they will be set to 0 and 0.05 respectively.

tacdf <- data.frame(GM = inptacdata$GM, 
                    ACC = inptacdata$gmACC, 
                    CER = inptacdata$gmCER, 
                    PUT = inptacdata$PUT)

Vndgrid <- seq(from=0, to=5, by=0.1)

SIMEout <-  SIME(t_tac, tacdf, input, Vndgrid, weights = weights, 
    inpshift = onetcmout$par$inpshift, vB = 0.05, twotcmstart = 1)

pandoc.table(SIMEout$par, digits=3)

Plot

The output object can be used to visualise the model fit:

plot_SIMEfit(SIMEout)

SIME Plot Fit Check

pandoc.table(summary(SIMEout))

Vndgrid <- seq(from=0, to=5, by=1)

SIMEout1 <-  SIME(t_tac, tacdf, input, Vndgrid, weights = weights, 
    inpshift = onetcmout$par$inpshift)

Vndgrid <- seq(from=SIMEout1$par$Vnd-1, to=SIMEout1$par$Vnd+1, length.out = 10)

SIMEout2 <-  SIME(t_tac, tacdf, input, Vndgrid, weights = weights, 
    inpshift = onetcmout$par$inpshift)

Vndgrid <- seq(from=SIMEout2$par$Vnd-0.2, to=SIMEout2$par$Vnd+0.2, length.out = 10)

SIMEout3 <-  SIME(t_tac, tacdf, input, Vndgrid, weights = weights, 
    inpshift = onetcmout$par$inpshift)